Finite Groups

List, to within isomorphism, all the finite groups whose orders do not exceed 5.

Show that all groups of order $\leq 5$ are commutative. Give an example of a noncommutative group of order 6.

Prove that any group of order 6 is isomorphic to either $\mathbb {Z}^{6}$ or $S_3$ (the group of permutations of three objects).

Find all abelian groups of order 8, up to isomorphism. Then identify which type occurs in each of

Show that there are at least two nonisomorphic nonabelian groups of order 24, of order 30 and order 40.

List eight groups of order 36 and prove that they are not isomorphic.

Prove that if $p$ is prime then every group of order $p^2$ is abelian.

Classify up to isomorphism all groups of order 45.

Let $G$ be a group of order 56 having at least 7 elements of order 7. Prove that $G$ has only one Sylow 2-subgroup $P$, and that all nonidentity elements of $P$ have order 2.

Classify all abelian groups of order 80 up to isomorphism.

Find (up to isomorphism) all groups of order $2p$, where $p$ is a prime ($p \geq 2$).

Prove that any finite group of order $n$ is isomorphic to a subgroup of $O(n)$, the group of $n \times n$ orthogonal real matrices.

Suppose that $G$ is a finite group such that every Sylow subgroup is normal and abelian. Show that $G$ is abelian.

If $G$ is a finite group, must $S = \left\{ g^2 \mid g \in G\right\}$ be a subgroup? Provide a proof or a counterexample.

Let $G$ be a finite group acting transitively on a set $X$ of size at least 2. Prove that some element $g$ of $G$ acts without fixed points.

Let $G$ be a finite non-Abelian group of order $n$. Show that there exists an integer $d$ satisfying $2 \leq d \leq n/2$ and a set $P$ of cardinality $d$, such that $G$ acts transitively on $P$.

Prove that every finite group is isomorphic to

Let $G$ be a finite group and $p$ a prime number. Suppose $a$ and $b$ are elements of $G$ of order $p$ such that $b$ is not in the subgroup generated by $a$. Prove that $G$ contains at least $p^2 - 1$ elements of order $p$.

Find all finite abelian groups $G$ (up to isomorphism) such that the group of automorphisms of $G$ has odd order.

Let $n$ be a positive integer. Let $\phi (n)$ be the Euler phi function, so $\phi (n) = \# (\mathbb {Z}^{n})^{*}$. Prove that if $\gcd (n, \phi (n)) > 1$, then there exists a noncyclic group of order $n$.